Farey Sequences Definition

Definition

A Farey sequence is a complete sequence of reduced form rational fractions in the range of the form with for . The sequence is complete in the sense that, given a Farey sequence as defined above, we can find no rational fraction with such that .

Theorem

Proof

Proof is by induction. For , we have the trivial Farey sequence where the above equality holds for . Given a Farey sequence , the sequence comprises a merge of and the sequence . If some is not in reduced form then it can be represented as a reduced form rational fraction with . But this fraction will already be a member of since is complete. Thus only reduced form fractions will be members of and will be merged with . The merge is such that all elements of are inserted in their correct position in the sequence in order to produce . (This observation enables an algorithm for the construction of for any to be designed.)

Proof

Lemma 3

Proof

This applies for all and can be applied successively to create a sequence .
But and hence for .
From Lemma 2 we have some such that are consecutive members of and where and are consecutive members of and is in reduced form.
Consider the rational fractions . We can prove the following lemma.

Lemma 4

Proof

Assume

which follows from Lemma 3. But Since is an integer, our original assumption is false and . Since , because there is no rational fraction such that as is complete.
The second equality can be proved using a similar technique

Assume

which follows from Lemma 3. But Since is an integer, our original assumption is false and . Since , because there is no rational fraction such that as is complete.
This proves Lemma 4.

From Lemmas 3 and 4, the proof of the theorem follows directly.

This shows that the rational fraction inserted into obeys the equality given in the Theorem. However, we still have to consider preceding values to check whether the Theorem is generally applicable to .

There are 2 possibilities:and where are consecutive terms in sequence .